Fundamentals of Plasma Physics

64 Chapter 3. Motion of a single plasma particle

Sinceηi(t 1 , 2 ) = 0the integrated term vanishes and sinceηiwas an arbitrary function oft,
the coefficient ofηiin the integrand must vanish, yieldingLagrange’s equation

dPi

dt

=

∂L

∂Qi

(3.7)

where thecanonical momentumPiis defined as

Pi=

∂L

∂Q ̇i

. (3.8)

Equation (3.7) shows that ifLdoesnotdepend on a particular generalized coordinate
QjthendPj/dt= 0in which case the canonical momentumPjis aconstant of the motion;
the coordinateQjis called acyclicorignorablecoordinate. This is a very powerful and
profound result. Saying that the Lagrangian function does not depend on a coordinate
is equivalent to saying that the system issymmetricin that coordinate or translationally
invariant with respect to that coordinate. The quantitiesPj andQjare called conjugate
and action has the dimensions of the product of these quantities.
Hamilton extended this formalism by introducing a new function related tothe La-
grangian. This new function, called the Hamiltonian, provides further usefulinformation
and is defined as

H≡

(

∑

i

PiQ ̇i

)

−L. (3.9)

Partial derivatives ofHwith respect toPiand toQigive Hamilton’s equations

Q ̇i=∂H

∂Pi

P ̇i=−∂H

∂Qi

(3.10)

which are equations of motion having a close relation to phase-space concepts.The time
derivative of the Hamiltonian is

dH

dt

=

∑

i

dPi

dt

Q ̇i+

∑

i

Pi

dQ ̇i

dt

−

(

∑

i

∂L

∂Qi

Q ̇+

∑

i

∂L

∂Q ̇i

dQ ̇i

dt

+

∂L

∂t

)

=−

∂L

∂t

.

(3.11)

This shows that ifLdoes not explicitly depend on time, i.e.,∂L/∂t= 0, the Hamiltonian
His aconstant of the motion.As will be shown later,Hcorresponds to the energy of the
system, so if∂L/∂t= 0,the energy is a constant of the motion. Thus, energy is conjugate
to time in analogy to canonical momentum being conjugate to position (note that energy×
time also has the units of action). If the Lagrangian does not explicitly dependon time, then
the system can be thought of as being ‘symmetric’ with respect to time, or ‘translationally’
invariant with respect to time.
The Lagrangian for a charged particle in an electromagnetic field is

L=

mv^2

2

+qv·A(x,t)−qφ(x,t); (3.12)

the validity of Eq.(3.12) will now be established by showing that it generatesthe Lorentz
equation when inserted into Lagrange’s equation. Since no symmetry is assumed, there is
no reason to use any special coordinate system and so ordinary Cartesian coordinates will